Counting of Figures Test - 8

The figure has been labelled as shown below:



The straight lines in the figure are as follows:
(i) Straight lines from the outer square: AB, BC, CD, AD = 4
(ii) Straight lines in the diagonals: AC, BD = 2
(iii) Other straight lines inside the square: EG, FH, EF, FG, GH, EH, IJ, KL = 8
Therefore, total number of straight lines is 14.

Number of straight lines in the given figure = 4 + 5 + 8 = 17
(IK, JE, IE, JF, KF, AB, AD, BC, DC, DH, HG, GC, HE, GF, JM, DE and FC)

The figure in the question may be marked at different points as shown in the following figure:



The triangles are as follows:
ABC, AFC, FBC, GHI, GKI, KHI, FED, FEI, FDI, FMN, FMK, FKN, PSC, PIC, ISC, ALF, GQM, NHR, JDS, EPL, FCJ, FCL, FRI, FQI, DRI, EQI and FBJ
Therefore, there are 27 triangles in the figure.

5 (LC, CF, FI, IL, LF) + 3 (BJ, CI, DH) + 6 (AE, EG, GK, KA, AG, EK) = 14 straight lines are required to make the given figure.

(Here, 1 element means 1 closed figure.)
There are 8 triangles that comprise 1 element each.
There are 5 triangles that comprise 2 elements each.
There are 3 triangles that comprise 3 elements each.
There is 1 triangle that comprises 4 elements.
There is 1 big triangle.
Thus, total number of triangles = 8 + 5 + 3 + 1 + 1 = 18
Hence, option 1 is correct.

We will label the figure as shown below:


The horizontal straight lines are BC, IL, FH and DE, i.e. 4.
The lines sloping to your left are IM, FN, DO and AC, i.e. 4.
The lines sloping to the right are LO, HN, EM and AB, i.e. 4.
Thus, the number of straight lines = 3 x 4 = 12
Thus, answer option 2 is correct.

The figure in the question may be labelled as shown below:

It is clear that there are five squares from the outermost square: ABCD, AGOE, GBFO, EOHD and OFCH.
Similarly, there is a total of 10 squares, five each from the middle and the innermost squares.
Hence, the total number of squares in the figure is 15.

The figure may be labelled as shown below.

There are 7 + 6 = 13 straight lines in the figure, namely AD, EH, IL, FG, BC, JK, LK, DC, HG, MN, EF, AB and IJ.



Thus, answer option 1 is correct.

There are a total of 19 squares in the figure, which have been marked as shown below:



There are three squares in the upper row – EFIH, FGJI and GBLJ.
There are five squares in the middle row – TQRS, QHPR, HIOP, IJNO and JLMN.
There are three squares in the lower row – SRUD, RPVU and POWV.
There are a total of six squares, three in each of the two rectangles (ABMS and TLCD): AEPS, EGNP, FBMO, THVD, QIWU and ILCW.
There are two squares in the main rectangle ABCD – AFWD and EBCV.
Hence, there are a total of 19 squares in the main figure.

We may label the figure as shown below:


The triangles are: ADF, AFB, BEF, DEF, DEC, BEC, ADB, DEB, AEB, ADE, FBC, DFC, ABC, ADC, DBC.
There are 15 triangles in the given figure.

The figure in the question may be labelled as shown below:


The triangles are: ELD, HLD, EDH, EIA, AIG, EAG, GJB, BJF, GBF, FKC, CKH, FCH, GOE, GOF, HOF, HOE, GEF, HEF, GEH and GHF.

There is a total of 20 triangles.

The figure may be marked as shown below.

There are 12 triangles comprising 1 element each.
There are 8 triangles comprising 2 elements each.
There are 4 triangles comprising 3 elements each.
There are 4 triangles comprising 6 elements each.
Thus, total number of triangles = 12 + 8 + 4 + 4 = 28

The figure may be labelled as shown.
The simplest triangles are PGI, PEM, REM, MHJ, JLS, SFO, FOQ and IKQ, i.e. 8 in number.
The triangles composed of two components each are EPR, QFS, AIE, EDJ, JCF, IBF, EIN, ENJ, FNJ and INF, i.e. 10 in number.
The triangles composed of four components each are EIF, IJF, EFJ and EIJ, i.e. 4 in number.
There are a total of 22 triangles in the figure.

The figure may be labelled as shown.
The simplest triangles are APQ, AEQ, QTU, QRU, BGS, BHS, RSU, SUV, TUW, UWX, NWD, WDM, UVY, UXY, JCY and YKC, i.e. 16 in number.
The triangles composed of two components each are QUW, QSU, SYU and UWY, i.e. 4 in number.
The triangles composed of three components each are AOU, AFU, FBU, BIU, UIC, ULC, ULD and OUD, i.e. 8 in number.
The triangles composed of four components each are QYW, QSW, QSY and SYW, i.e. 4 in number.
The triangles composed of six components each are AUD, ABU, BUC and DUC, i.e. 4 in number.
The triangles composed of seven components each are QMC, ANY, EBW, PSD, CQH, AGY, DSK and BJW, i.e. 8 in number.
The triangles composed of twelve components each are ABD, ABC, BCD and ACD, i.e. 4 in number.
Thus, total number of triangles in the figure = 16 + 4 + 8 + 4 + 4 + 8 + 4 = 48

The figure in question may be labelled as shown below.



AEKI, EGJK, AGJI, IKOL, KJMO, IJML, LORP, OMQR, LMQP, PRFD, RQHF, PQHD - 12
AGML, EBNO, ABNL, IJQP, LMHD, ONCF, LNCB - 7
APRE, IKFD, AEFD, ERQG, KFHJ, EFHG, GBCH - 7
There is one more rectangle ABCD.
Hence, there are a total of 27 rectangles.

We label the figure as shown.
The simplest triangles are BFG, CGH, EFM, FMG, GMN, GHN, HNI, LMK, MNK and KNJ, i.e, 10 in number.
The triangles composed of three components each are FAK and HKD, i.e, 2 in number.
The triangles composed of four components each are BEN, CMI, GLJ and FHK, i.e, 4 in number.
The triangles composed of eight components each are BAJ and CLD, i.e, 2 in number.
Thus, there are 10 + 2 + 4 + 2 = 18 triangles in the given figure.

The figure in the question may be labelled as shown below.



There are a total of 14 squares in the given figure.
Between the lines RJ and OK there are 4 squares, between lines EI and RS there are 3 squares,
between lines DG and EH there are 2 squares and between lines AB and CD there is 1 square.
4 more squares are there (RFQN, DGQR, EHMO and FNLI).
Total number of squares in the figure are (4 + 3 + 2 + 1 + 4) = 14.

AHB, GFE, GIE, IJE, EJC, CED, GHI, BJC = 8
HBE = 1
GHE, BCE, IEC, GJE = 4
FHE, EBD, GEC, AGC = 4
AFD = 1
Total number of triangles = 18

The figure may be marked as shown below.



There are 10 triangles comprising 1 element each.
There are 7 triangles comprising 2 elements each.
There are 4 triangles comprising 3 elements each.
There are 2 triangles comprising 4 elements each.
There is 1 triangle comprising 6 elements.
Thus, total number of triangles = 10 + 7 + 4 + 2 + 1 = 24
Hence, answer option (3) is correct.

The figure may be labelled as shown.



There are 9 triangles comprised of 1 element each.
There are 9 triangles comprised of 2 elements each.
There is 1 triangle comprised of 4 elements.
There are 4 triangles comprised of 5 elements each.
There are 2 triangles comprised of 6 elements each.
There are 2 big triangles.
Thus, total number of triangles = 9 + 9 + 1 + 4 + 2 + 2 = 27.
Hence, answer option 4 is correct.

There are eight triangles in the upper half of the figure, namely ABE, AEO, ABO, AOF, AFC, AOC, AEF and ABC.



Similarly, there are eight triangles in the lower half of the figure.
There are eight more triangles, namely ABD, AED, AFD ,ACD , ABF, ACE, DBF and DCE.
Hence, there are total 24 triangles in the figure.

The figure may be labelled as shown.



AGH, AIG, AIB, FJE, JCE, CED (6)
ABG, FCE, ACE, AGE (4)
IGE, CFD, AJC (3)
AHE (1)
So, there are 6 + 4 + 3 + 1 = 14 triangles in the given figure.

There are 5 vertical, 5 horizontal and 2 diagonal lines, i.e. a total of 12 straight lines.

The figure may be labelled as shown:
The number of quadrilaterals in the figure is 11.



ABCD, AFED, AGDH, ACDF, ABDF, EACD, CAHD, ABDE, AGDF, ABDH and EAGD.

The pentagons in the figure are ABCDE, ABCDF, ABCDH, ABDEF, CDEFA, DEFAG and DEFAH. Clearly, there are seven pentagons in the figure.
Hence, option (4) is the correct answer.